Math’s AssignMent

Areas with Combined of Other sides

Area Area is the quantity that expresses the extent of a two-dimensional figure or 

shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.[1] It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size.[2] In the International System of Units (SI), the standard unit of area is the square meter (written as m2), which is the area of a square whose sides are one meter long.[3] A shape with an area of three square meters would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, 

rectangles, and circles. Using these formulas, the area of any polygon can be found by 

dividing the polygon into triangles.[4] For shapes with curved boundary, calculus is usually

required to compute the area. Indeed, the problem of determining the area of plane figures was

a major motivation for the historical development of calculus.[5]

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called

the surface area.[1][6] Formulas for the surface areas of simple shapes were computed by the 

ancient Greeks, but computing the surface area of a more complicated shape usually requires 

multivariable calculus.

Circle area In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the

region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature

 of the lune of Hippocrates,[12] but did not identify the constant of proportionality. 

Eudoxus of Cnidus, also in the 5th century BCE, also found that the area of a disk is proportional

to its radius squared.[13] Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle.

Triangle area Heron (or Hero) of Alexandria found what is known as Heron's formula for the area of a triangle

in terms of its sides, and a proof can be found in his book, Metrica, written around 60 CE. It has

been suggested that Archimedes knew the formula over two centuries earlier,[15] and

since Metrica is a collection of the mathematical knowledge available in the ancient world, it is

possible that the formula predates the reference given in that work.[16]

In 499 Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, expressed the area of a triangle as one-half the base times the height in theAryabhatiya 

Quadrilateral area In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula,

for the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In

1842 the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt

 independently found a formula, known as Bretschneider's formula, for the area of any

quadrilateral.

General polygon area The development of Cartesian coordinates by René Descartes in the 17th

century allowed the development of the surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century.

Presented byR.Sanjith Varun IX - B